1. Chapter 15
Oscillations

2. Periodic motion Periodic (harmonic) motion – self-repeating motion
Oscillation – periodic motion in certain direction
Period (T) – a time duration of one oscillation
Frequency (f) – the number of oscillations per unit time, SI unit of frequency 1/s = Hz (Hertz)
Heinrich Hertz
( )

3. Simple harmonic motion
Simple harmonic motion – motion that repeats itself and the displacement is a sinusoidal function of time

4. Amplitude
Amplitude – the magnitude of the maximum displacement (in either direction)

5. Phase

6. Phase constant

7. Angular frequency

8. Period

9. Velocity of simple harmonic motion

10. Acceleration of simple harmonic motion

11. The force law for simple harmonic motion
From the Newton’s Second Law:
For simple harmonic motion, the force is proportional to the displacement
Hooke’s law:

12. Chapter 15
Problem 16

13. Energy in simple harmonic motion
Potential energy of a spring:
Kinetic energy of a mass:

14. Energy in simple harmonic motion

15. Chapter 15
Problem 37

16. Pendulums Simple pendulum: Restoring torque:
From the Newton’s Second Law:
For small angles

17. Pendulums
Simple pendulum:
On the other hand

18. Pendulums
Simple pendulum:

19. Pendulums
Physical pendulum:

20. Chapter 15
Problem 51

21. Simple harmonic motion and uniform circular motion
Simple harmonic motion is the projection of uniform circular motion on the diameter of the circle in which the circular motion occurs

22. Simple harmonic motion and uniform circular motion
Simple harmonic motion is the projection of uniform circular motion on the diameter of the circle in which the circular motion occurs

23. Simple harmonic motion and uniform circular motion
Simple harmonic motion is the projection of uniform circular motion on the diameter of the circle in which the circular motion occurs

24. Damped simple harmonic motion
Damping
force
Damping
constant

25. Forced oscillations and resonance
Swinging without outside help – free oscillations
Swinging with outside help – forced oscillations
If ωd is a frequency of a driving force, then forced oscillations can be described by:
Resonance:

26. Answers to the even-numbered problems
Chapter 15:
Problem 2
10 N;
(b) 1.2 × 102 N/m

27. Answers to the even-numbered problems
Chapter 15:
Problem 28
200 N/m;
(b) 1.39 kg;
(c) 1.91 Hz

28. Answers to the even-numbered problems
Chapter 15:
Problem 38
12 s

29. Answers to the even-numbered problems
Chapter 15:
Problem 42
0.499 m;
(b) mJ

30. Answers to the even-numbered problems
Chapter 15:
Problem 58
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